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**Local adaptive observer for linear time-varying systems** **with parameter-dependent state matrices**

### Romain Postoyan, Qinghua Zhang

**To cite this version:**

### Romain Postoyan, Qinghua Zhang. Local adaptive observer for linear time-varying systems with

### parameter-dependent state matrices. 2018. �hal-01869339�

### Local adaptive observer for linear time-varying systems with parameter-dependent state matrices

Romain Postoyan and Qinghua Zhang

Abstract— We present an adaptive observer for linear time- varying systems whose state matrix depends on unknown parameters. We first assume that the state matrix is affine in these parameters. In this case, the proposed observer generates state and parameter estimates, which exponentially converge to the plant state and the true parameter, respectively, provided a persistence of excitation condition holds and the unknown parameters lie in a neighborhood of some known nominal values. Hence, some prior knowledge on the unknown parameters is required, but not on the state. We then modify the adaptive observer and its convergence analysis to systems whose state matrix is smooth, instead of being affine, in the unknown parameters. The convergence is approximate, and no longer exponential, in this case. An example is provided to il- lustrate the results, for which the required distance between the unknown parameters and their nominal values is investigated numerically.

I. INTRODUCTION

The objective of this work is to estimate both the state and the parameters of the system

˙

x(t) = A(t, θ)x(t) +B(t)u(t)

y(t) = C(t)x(t), (1)

where^{1} x(t) ∈ R^{n}^{x} is the state, u(t) ∈ R^{n}^{u} is the
input, y(t) ∈ R^{n}^{y} is the output, θ ∈ R^{n}^{θ} is the vector
of unknown constant parameters, t ≥ 0 is the time, and
nx, nu, ny, nθ ∈Z^{>0}. In other words, we aim at designing
an adaptive observer for system (1). This problem arises
in many practical applications, including electrochemical
batteries [13] or mechanical systems [9] to cite a couple
of examples, as soon as we need to estimate on-line both
the unmeasured state and unknown parameters of a plant for
monitoring purposes for instance.

In the majority of works on adaptive observers, the state matrix is assumed to be fully known contrary to (1), and the unknown parameters enter in the system through an additional term of the form Φ(t, u(t), y(t))θ where Φ is known. Solutions for this case can be found in e.g., [3], [4], [20], [28]–[30]. The fact thatAdepends onθin (1) is a major difference with these references, which makes the problem difficult because of the cross-terms involving θ and x(t).

There are nevertheless results in the literature, which propose adaptive observers for systems with partially unknown state

R. Postoyan is with the Universit´e de Lorraine, CNRS, CRAN, F-54000 Nancy, Franceromain.postoyan@univ-lorraine.fr.

Q. Zhang is with Inria-IFSTTAR, Campus de Beaulieu, 35042 Rennes Cedex, Franceqinghua.zhang@inria.fr.

This report corresponds to [23] with all the proofs as well as further details.

1The notation and the terminology are defined at the end of the introduc- tion.

matrixA. In [17], several designs are given for single-input single-output linear time-invariant (LTI) systems. In [18], it is explained how to optimally fit a discrete-time LTI model to any given output signals. In [21], a switched approach is proposed for LTI systems based on linear matrix inequalities (LMI) conditions. More recently, a generic approach for the state and parameter estimation of nonlinear systems was presented in [8] using a multi-observer set-up. On the other hand, adaptive observers for systems with known state matrices with an additive term of the form Φ(x)θ(see e.g., [5], [7], [11], [12]) are relevant in this context when A is affine in θ, as a simple manipulation brings system (1) to the right form.

In this work, we propose an approach, which relies on a simple idea. We essentially assume that we know a state- observer for system (1) for some known nominal param- eter value θ0. Because θ 6= θ0, the idea is to modify the original state-observer and design an adaptation law to estimate θ by relying on similar techniques as in [30].

Indeed, by adding and subtracting A(t, θ0)x(t) in the first
equation in (1), we obtainx(t) =˙ A(t, θ0)x(t) +B(t)u(t) +
[A(t, θ)−A(t, θ0)]x(t). Assuming A(t, θ)is affine inθ we
can write A(t, θ)x(t)−A(t, θ_{0})x(t) = Λ(t, x(t))(θ−θ_{0}),
whereΛ(t, x(t))is a time-varying matrix linear in its second
argument; this is explained in more detailed in the sequel.

Hence,x(t) =˙ A(t, θ_{0})x(t) +B(t)u(t) + Λ(t, x(t))(θ−θ_{0}).

Denoting by xˆ the state estimate of the adaptive observer
to be designed and x˜ = x−xˆ the associated state esti-
mation error, we derive x(t) =˙ A(t, θ0)x(t) +B(t)u(t) +
Λ(t,x(t))(θˆ −θ0) + Λ(t,x(t))(θ˜ −θ0), sinceΛ is linear in
its second argument. NowΛ(t,x(t))ˆ is known sincexˆis the
state estimate we generate. The term Λ(t,x(t))(θˆ −θ0) is
therefore of the formΦ(t)(θ−θ0)whereΦ(t)is known. We
are almost back to the same class of systems as in [30] where
the unknown parameter isθ−θ0instead ofθ, but this is fine
since we knowθ0. The difference with [30] is that we have
to deal with the extra term Λ(t,x(t))(θ˜ −θ0). It appears
that when ||θ−θ_{0}|| is small, so is Λ(t,x(t))(θ˜ −θ_{0})˜x(t)
so that stability of the estimation error may be ensured.

The point of these developments is to show that after some simple manipulations, we can write the problem in a similar, though different form, as in [30]. Still, the design of the adaptive observer is not a straightforward application of [30]

and requires appropriate modifications as well as specific assumptions. The exponential convergence of the proposed adaptive observer is guaranteed provided a persistence of excitation condition holds and ||θ−θ0|| is small enough.

Such an excitation condition is needed in almost all works

on adaptive observers. The fact that||θ−θ0||has to be small means that the design is local in the parameter estimate, but we are free to initialize the state estimate as we wish. It can be noted that many estimation or identification schemes are also local, see [18] for instance and all the works on extended Kalman filters, see e.g., [6], [16], [24]. Afterwards, the results are extended to the case where Ais only smooth in θ, and not necessarily affine. The adaptive observer and its analysis are modified accordingly and an approximate convergence property is ensured in this case.

Compared to [17], our results apply to multi-input multi-
output linear time-varying systems, as opposed to single-
input single-output LTI systems, and compared to [21], we
address LTV systems and our approach relies on different
type of conditions (no LMI). The closest work in spirit,
although it is written in discrete-time for LTI models, is
the one in [18]. A key difference with [18] is that we
do not need to use projection techniques to maintain the
parameter estimate in some given set, and the estimates
we generate cannot be trapped in the border of the given
set due to projections, but always converge to the true
parameter provided, again, that ||θ−θ_{0}|| is small enough
and a persistence of excitation holds. On the other hand, the
adaptive observer is less computationally demanding than the
supervisory observer presented in [8], which requires a large
number of observers to work, and we ensure the asymptotic
convergence of the estimation errors as opposed to a practical
property in [8]. Finally, compared to [5], [7], [11], [12]: (i)
we do not require any particular triangular or block-diagonal
structure of the system ([5], [11], [12]); (ii) we provide
results for the case where the state matrix is not affine in θ
in which case the results in [5], [7], [11] do not apply; (iii)
we do not rely on high-gain techniques ([11], [12]), which
are not known to be very sensitive to noise measurements;

(iv) the observer is of smaller dimension, which is relevant for computational reasons ([5], [11], [12]). Also, we believe that the way we approach the problem has its own interest, which can be relevant for further extensions.

A similar problem is addressed in [9]. First, exponential convergence is ensured in our case whenAis affine inθ as opposed to an ultimate boundedness property in [9]. Second, the proposed solution is computationally lighter, since [9]

requires the use of both a state-observer for the nominal parameter values and of an adaptive observer, while we only need the latter.

The rest of the paper is organized as follows. The assump- tions we make on system (1) are presented in Section II. The main result is then given in Section III and an illustrative example is provided in Section IV. Section V concludes the paper. A technical lemma is provided in the appendix.

Notation and terminology. Let R := (−∞,∞), R≥0 :=

[0,∞), R^{>0} := (0,∞), Z≥0 :={0,1,2, . . .}, and Z^{>0} :=

{1,2, . . .}. The notation(x, y)stands for [x^{>}, y^{>}]^{>}, where
x∈R^{n},y ∈ R^{m}, and n, m∈Z>0. The identity matrix of
sizen∈Z>0is denotedIn or simplyIwhen the dimension
is clear from the context. Given two matricesQ, R∈R^{n×n},
we writeQ≤RifR−Qis positive semi-definite,n∈Z>0.

The symbol ? stands for symmetric blocks in matrices. A
time-varying matrix M(t) ∈ R^{n×m} is said to be bounded
if there exists c ≥0 such that ||M(t)|| ≤c for anyt ≥0,
where||M(t)|| := sup_{x∈}_{R}m\{0}||M(t)x||

||x|| , || · ||standing for
the Euclidean norm when the argument is a vector. We write
N(x) = O(||x||^{2}) for N : R^{n} → R^{m} with n, m ∈ Z>0,
when there exists c ≥ 0 such that ||N(x)|| ≤ c||x||^{2} for
anyx∈R^{n}. The set of piecewise continuous functions from
R≥0 to R^{n} is denoted PC(R^{n}),n∈Z>0. Let p∈R^{n} and
δ∈R>0∪ {∞}, the closed ball of radiusδ centered atpis
denotedB(p, δ), withB(p, δ) =R^{n} whenδ=∞.

II. ASSUMPTIONS

The matrices A(t, θ), B(t) andC(t) in (1) are assumed to be continuous and bounded with respect to the timet. We make the next assumption on howA(t, θ)depends onθ.

Assumption 1: The matrix A(t, θ) is affine in θ, in the
sense that there exist matricesA0(t), . . . , An_{θ}(t)∈R^{n}^{x}^{×n}^{x}
such that, for any t ≥0, A(t, θ) =A0(t) +Pn_{θ}

i=1Ai(t)θi,

whereθ= (θ1, . . . , θn_{θ}).

When Assumption 1 is not satisfied, we may redefine the
vector of unknown parameters to enforce it, by eventually
over-parameterizing the original system. This assumption
will be relaxed in Section III-C. It is important to note
that the matrices A_{0}(t), . . . , A_{n}_{θ}(t) in Assumption 1 are
known, as these can be directly derived from the expression
ofA(t, θ). These matrices are continuous and bounded with
respect to the time since so isA(t, θ).

Our design implicitly relies on the knowledge of a state observer for system (1) ifθwould be known. More precisely, we make the following assumption.

Assumption 2: There exist δ1 ∈ R^{>0}∪ {∞} and K :
R≥0 × B(θ, δ1) → R^{n}^{x}^{×n}^{y}, which is continuous and
bounded, such that for any θ0 ∈ B(θ, δ1) the origin of the
system x(t) = [A(t, θ˙˜ 0)−K(t, θ0)C(t)] ˜x(t) is uniformly

globally exponentially stable.

Assumption 2 essentially means that we know a set where
parameter θ lies and, if we knew θ, we would be able to
design a state-observer for the corresponding system. More
precisely, Assumption 2 states that there exists a neigh-
borhood of parameter θ, corresponding to B(θ, δ1), such
that we can design a standard (Luenberger) state observer
with gain K(t, θ_{0}) for any θ_{0} in this set, whose solutions
uniformly, globally, and exponentially converge to solutions
to (1) when θ = θ_{0}. To see it, let x(t) =˙ˆ A(t, θ_{0})x(t) +
B(t)u(t) + K(t, θ_{0})(y(t) −C(t)ˆx(t)). If θ = θ_{0}, then
the estimation error system with variable x˜ = x−xˆ is

˙˜

x(t) = [A(t, θ0)−K(t, θ0)C(t)] ˜x(t)as in Assumption 2.

A necessary and sufficient condition for Assumption 2 to hold is provided next.

Lemma 1: Under Assumption 1, Assumption 2 holds if
and only if there exists a continuous and bounded mapping
K^{∗} : R≥0 → R^{n}^{x}^{×n}^{y} such that the origin of x(t) =˙˜

[A(t, θ)−K^{∗}(t)C(t)] ˜x(t) is uniformly globally exponen-
tially stable for the (unknown) true parameter valueθ.

Proof. The necessity part immediately follows from As- sumption 2. For the sufficiency, we consider the func-

tion W(P^{∗},x)˜ := x˜^{>}P^{∗}x˜ where P^{∗} is the solution
to −P˙^{∗}(t) = P^{∗}(t)

A(t, θ) −K^{∗}(t)C(t)
+

A(t, θ) −
K^{∗}(t)C(t)>

P^{∗}(t) +I. According to Theorem 4.12 in [15],
such a matrixP(·)exists onR≥0, is continuously differen-
tiable, bounded, symmetric, positive definite and there exist
a_{P}∗, aP^{∗} > 0 such that a_{P}∗I ≤ P^{∗}(t) ≤ aP^{∗}I for any
t ≥ 0. Let θ0 ∈ R^{n}^{θ} and t ≥ 0. Along the solutions
to x(t) = [A(t, θ˙˜ 0)−K^{∗}(t)C(t)] ˜x(t), W˙ (P^{∗}(t),x(t)) =˜

−||˜x(t)||^{2} + 2˜x(t)^{>}P^{∗}(t) [A(t, θ0)−A(t, θ)] ˜x(t). Since
A(t, θ)is bounded intand affine inθaccording to Assump-
tion 1, there existsδ_{1}>0such that for anyθ_{0}such that||θ0−
θ|| < δ_{1}, W˙ (P^{∗}(t),x(t))˜ ≤ −^{1}_{2}||˜x(t)||^{2}. Consequently, the
origin of x(t) = [A(t, θ˙˜ 0)−K^{∗}(t)C(t)] ˜x(t) is uniformly
globally exponentially stable according to Theorem 4.10 in
[15]. Hence, Assumption 2 holds withK(t, θ0) =K^{∗}(t).

Lemma 1 involvesθ, which we do not know. The follow- ing lemma provides exploitable conditions for the satisfaction of Assumption 2.

Lemma 2: Suppose the following holds.

(i) Assumption 1 holds.

(ii) There exists δ_{1} ∈ R≥0 ∪ {∞} such that the pair
(A(t, θ_{0}), C(t, θ_{0})) is uniformly completely observ-
able^{2} for any θ0∈ B(θ, δ1).

Assumption 2 is verified with K(t, θ_{0}) =
P(t, θ_{0})C(t)^{>}R^{−1}(t) where P(t, θ_{0}) is the solution
to P˙(t, θ_{0}) = P(t, θ_{0})A(t, θ_{0})^{>} + A(t, θ_{0})P(t, θ_{0}) −
P(t, θ_{0})C(t)^{>}R^{−1}(t)C(t)P(t, θ_{0}) +Q(t)withP(0) =P_{0},
where P_{0} is any symmetric, positive definite matrix, and
R(t), Q(t) are any symmetric, positive definite, continuous

and bounded matrices.

Proof. Let θ_{0} ∈ B(θ, δ_{1}). Since R(t) is positive definite
and the pair(A(t, θ0), C(t, θ0))is uniformly completely ob-
servable, the pair(A(t, θ0), C(t, θ0)R^{1}^{2}(t))is also uniformly
completely observable. Moreover, the matrix W(t0, t) in
[1] is nonsingular at t0 sinceP0 is symmetric and positive
definite, it is therefore nonsingular at somet1> t0by conti-
nuity. Consequently, all the conditions of Theorem 3.1 in [1]

are verified, which ensures the uniform global exponential
stability of the origin ofx(t) = [A(t, θ)˙˜ −K^{∗}(t)C(t)] ˜x(t).

SinceA(t, θ0)is continuous inθ0 in view of Assumption 1, so isP(t, θ0) according to Corollary 6 in Chapter 2 in [2].

Therefore, K(t, θ_{0}) is continuous in θ_{0} for θ_{0} ∈ B(θ, δ1).

Boundedness of K(t, θ_{0})follows from the boundedness of
P(t, θ_{0}), which is ensured in [1]. We have proved that

Assumption 2 is satisfied.

When the matrices A and C in (1) are time-invariant,
it suffices to have the pair (A(θ_{0}), C(θ_{0})) observable for
any θ_{0} with ||θ−θ_{0}|| ≤ δ_{1} for some δ_{1} ∈ R>0 ∪ {∞}

to ensure Assumption 2. Indeed, in this case, Bass-Gura
formula^{3} leads to a gain K(θ0), which is continuous inθ0.
The same applies when L(θ0) is selected to minimize a
linear quadratic cost, i.e. whenL(θ0) = P(θ0)C^{>}R^{−1} and
P(θ0)is the solution to the Riccati equationP(θ0)A(θ0)^{>}+

2See [26] for a definition of uniform complete observability as well as conditions to ensure it.

3See Chapter 4 in [14] for instance.

A(θ0)P(θ0)−P(θ0)C^{>}R^{−1}CP(θ0)+Q. It suffices to select
Q and R symmetric and positive definite to ensure the
continuity ofP(θ_{0})inθ_{0} according to Proposition 1 in [10],
which in turn implies the continuity ofK(θ_{0})inθ_{0}.

We finally make the next boundedness assumption on system (1).

Assumption 3: There exists a hyper-rectangle X ={x=
(x_{1}, . . . , x_{n}_{x}) ∈ R^{n}^{x} : x_{i} ∈ [c_{i}, c_{i}]} where c_{i} ≤ c_{i} ∈ R,
i∈ {1, . . . , n_{x}}, a non-empty set of inputsM_{u}⊂ PC(R^{n}^{u}),
and a non-empty set of initial conditions S_{X} ⊂R^{n}^{x}, such
that any solutionxto system (1) initialized inS_{X} with input
u∈ Mu satisfiesx(t)∈ X for allt≥0.

Assumption 3 means that the solutions to (1) lie in a known compact set, which can always be embedded in a closed hyper-rectangle, whenever its input lies in set Mu. Note that there is no restriction of the “size” ofX.

III. MAIN RESULT

A. Design and analysis

Like in [25], we introduce the element-wise satura-
tion function σ_{X} : R^{n}^{x} → R^{n}^{x} defined as σ_{X}(x) :=

(σ_{1,X}(x_{1}), . . . , σ_{n}_{x}_{,X}(x_{n}_{x}))withσ_{i,X}(x_{i}) =x_{i} when x_{i}∈
[c_{i}, ci], σ_{i,X}(xi) = c_{i} when xi ≤ c_{i} and σ_{i,X}(xi) = ci

when xi ≥ci, for any xi ∈ R, where X, c_{i}, ci come from
Assumption 3,i∈ {1, . . . , nx}. Based on Assumptions 1-2
and inspired by [30], we propose the adaptive observer

˙ˆ

x(t) = A(t, θ0)ˆx(t) +B(t)u(t) + Λ(t, σ_{X}(ˆx(t)))¯θ(t)
+

K(t, θ0) +γΥ(t)Υ^{>}(t)C^{>}(t)Σ(t)

×(y(t)−C(t)ˆx(t))

θ(t)˙¯ = γΥ^{>}(t)C^{>}(t)Σ(t)(y(t)−C(t)ˆx(t))

Υ(t)˙ = [A(t, θ_{0})−K(t, θ_{0})C(t)] Υ(t) + Λ(t, σ_{X}(ˆx(t)))
θ(t)ˆ = θ(t) +¯ θ_{0},

(2)
where θ(t)¯ ∈ R^{n}^{θ} is the estimate of θ − θ0, θ(t)ˆ ∈
R^{n}^{θ} is therefore the estimate of θ, Υ(t) ∈ R^{n}^{x}^{×n}^{θ},
Λ(t, z) := [A1(t)z . . . An_{θ}(t)z] ∈ R^{n}^{x}^{×n}^{θ} for z ∈ R^{n}^{x}
withA_{1}(t), . . . , A_{n}_{θ}(t) coming from Assumption 1, Σ(t) :
R→R^{n}^{y}^{×n}^{y} is any symmetric, positive definite, bounded,
continuous matrix,γ∈R>0, and X ⊂R^{n} comes from As-
sumption 3. We have two degree of freedom when designing
(2):Σ(t)and γ. The former has to be designed to verify a
persistence of excitation condition stated next. In practice, we
typically take it diagonal. Parameterγtunes the convergence
speed ofθ, the bigger¯ γ, the faster the convergence, but the
more sensitive to noise.

Remark 1: The use of the saturation function σX in (2) is essential to guarantee the boundedness ofΥin the proof of the next theorem, which is needed to establish the desired convergence property. This is the main difference with [30] in terms of design. Other differences reside in the assumptions

we rely on.

The next theorem ensures that the state and the parameter estimates provided by (2) exponentially converges toxand θ, provided a persistence of condition holds and||θ−θ0||is sufficiently small. Its proof is provided in Section III-B.

Theorem 1: Consider systems (1) and (2) and suppose the following holds.

(i) Assumptions 1-3 hold.

(ii) Let δ_{2} ∈ R>0 ∪ {∞}, xˆ_{0} ∈ R^{n}^{x}, θ¯_{0} ∈ R^{n}^{θ},
Υ_{0} ∈ R^{n}^{x}^{×n}^{θ}, there exist α, T > 0, which depend
onxˆ_{0},θ¯_{0},Υ_{0}, such that, for anyx(0)∈ SX and input
u∈ M_{u}, anyθ_{0} such that||θ−θ_{0}|| ≤δ_{2}, the solution
ˆ

x,θ,¯ Υ to system (2) initialized at (ˆx_{0},θ¯_{0},Υ_{0}) with

||θ_{0}−θ|| ≤δ_{2}, verifies for anyt≥0,
αI≤

Z t+T

t

Υ^{>}(τ)C^{>}(τ)Σ(τ)C(τ)Υ(τ)dτ. (3)
There exist δ ∈ (0,min{δ_{1}, δ_{2}}) and c_{1}, c_{2} > 0 , which
depend on X,M_{u},xˆ_{0},θ¯_{0},Υ_{0}, such that, if ||θ−θ_{0}|| ≤ δ,
the solution(ˆx,θ,¯ Υ)to (2) initialized at(ˆx_{0},θ¯_{0},Υ_{0})and any
solution to system (1) withx(0)∈ S_{X} and input u∈ Mu,
verify||(x(t)−x(t),ˆ θ(t)−θ)|| ≤ˆ c_{1}e^{−c}^{2}^{t}||(x(0)−xˆ_{0},θ(0)ˆ −

θ)|| for anyt≥0.

Item (ii) of Theorem 1 is a persistence of excitation
condition on the Υ-system, like in [30]. It actually is a
uniform persistence of excitation property, see Definition 2
in [19], as that the constants α and T are independent of
θ_{0}, which is essential. Indeed, if the bound δ_{2}, and thus
δ, on the norm of θ−θ_{0} would depend on θ_{0}, there will
be no guarantee that we can always select θ_{0} sufficiently
close to θ such that ||θ−θ_{0}|| ≤δ. That is also the reason
why we assume the existence of the gain K(t, θ_{0})for any
θ_{0} ∈ B(θ, δ_{1}) in Assumption 2, and not for a given θ_{0},
otherwise δ1 in Assumption 2 may depend on θ0 and the
same issue would arise. On the other hand, δ depends on
the initial conditions of observer (2). It is possible to ensure
thatδis uniform over these initial conditions over given sets,
when (3) holds for any initial conditions in these sets.

Property (3) is difficult to verify as it involves anyθ_{0}in a
set, and any solution initialized in SX with input u∈ Mu.
Even if we could verify it, that would not mean that the
estimates converge to the true values, as we also need for
that ||θ−θ_{0}|| to be sufficiently small, which is something
we cannot check in practice. It is important to note that
this difficulty is generic to any local estimation schemes,
including works on extended Kalman filters where the initial
conditionsx0andxˆ0 have to be close to each other (among
other conditions), see e.g., [6], [16], [24]. The only way to
avoid it is when we have the property that the observation
error y(t)−C(t)ˆx(t) converging to zero implies that the
estimation errors x(t)−x(t)ˆ and θ−θ(t)ˆ also converging
to zero, see Section III.A in [22]. In practice, we can check
numerically on-line whether the inequality in (3) holds for
a givenθ0 and a given solutionxto (1), in which case the
non-satisfaction of (3) indicates that the obtained estimate
are not valid.

The fact that we need to know a set where the unknown parameters lie is a valid assumption in various applications, like electrochemical batteries for which the parameters are accurately known when the battery is produced, but slowly vary with time. Then, the interest of the adaptive observer is to track these slow variations for monitoring purposes. If we

only have some vague knowledge aboutθ, we may overcome this issue by employing a multi-observer architecture as explained in [22].

Remark 2: Observer (2) can easily be shown to be robust
in the following sense, under the conditions of Theorem
1. Consider system (1) perturbed as x(t) =˙ A(t, θ)x(t) +
B(t)u(t) +w_{x}(t), y(t) = C(t)x(t) +w_{z}(t), where w_{x}
and w_{z} are disturbances and noise signals respectively.

The Lyapunov analysis in Section III-B almost immediately allows to conclude that the estimation error system satisfies an input-to-state stability property [27] in this case.

provided the conditions of Theorem 1 hold.

B. Proof of Theorem 1

We introduce the estimation errors x˜ :=x−xˆ and θ˜:=

θ−θ0−θ. Let¯ x(0) ∈ R^{n}^{x}, t ≥0 for which the solution
to (1), (2) is defined, and θ0 ∈ B(θ,min{δ1, δ2}) where
δ1, δ2 come from Assumptions 2 and item (ii) of Theorem
1, respectively. In view of (1) and (2),

˙˜

x(t) = A(t, θ)x(t) +B(t)u(t)−A(t, θ_{0})ˆx(t)−B(t)u(t)

−Λ(t, σ_{X}(ˆx(t)))¯θ(t)−[K(t, θ_{0})
+γΥ(t)Υ^{>}(t)C^{>}(t)Σ(t)

(y(t)−C(t)ˆx(t)), (4) we add and subtractA(t, θ0)x(t)and we obtain

˙˜

x(t)=[A(t, θ0)−K(t, θ0)C(t)] ˜x(t)

+ [A(t, θ)−A(t, θ0)]x(t)−Λ(t, σX(ˆx(t)))¯θ(t)

−γΥ(t)Υ^{>}(t)C^{>}(t)Σ(t)(y(t)−C(t)ˆx(t)).

(5)

In view of Assumption 1 and the definition ofΛ(·,·)given after (2),

[A(t, θ)−A(t, θ0)]x(t) = Λ(t, x(t))(θ−θ0). (6)
We now restrict our attention to the case wherex(0)∈ S_{X}
withu∈ Mu. Hence, x(t)∈ SX and x(t) =σX(x(t))in
view of Assumption 3 and the definition ofσ, respectively.

Thus [A(t, θ)−A(t, θ0)]x(t) = Λ(t, σX(x(t)))(θ − θ0).

Consequently, in view of (5),

˙˜

x(t)=[A(t, θ0)−K(t, θ0)C(t)] ˜x(t)

+Λ(t, σ_{X}(x(t)))(θ−θ0)−Λ(t, σ_{X}(ˆx(t)))¯θ(t)

−γΥ(t)Υ^{>}(t)C^{>}(t)Σ(t)(y(t)−C(t)ˆx(t)).

(7)

We add and subtractΛ(t, σ_{X}(ˆx(t)))(θ−θ0)and we obtain,
by exploiting the linearity ofΛ(·,·) in its second argument
and usingθ(t) =˜ θ−θ0−θ(t),¯

˙˜

x(t)=[A(t, θ0)−K(t, θ0)C(t)] ˜x(t)
+Λ(t, σX(x(t))−σX(ˆx(t)))(θ−θ0)
+Λ(t, σ_{X}(ˆx(t)))˜θ(t)

−γΥ(t)Υ^{>}(t)C^{>}(t)Σ(t)(y(t)−C(t)ˆx(t)).

(8)

Like in [30], we introduceη := ˜x−Υ˜θ. We deduce from

the last equation and (2),

˙

η(t)=[A(t, θ_{0})−K(t, θ_{0})C(t)] ˜x(t)
+Λ(t, σ_{X}(x(t))−σ_{X}(ˆx(t)))(θ−θ_{0})
+Λ(t, σ_{X}(ˆx(t)))˜θ(t)

−γΥ(t)Υ^{>}(t)C^{>}(t)Σ(t)(y(t)−C(t)ˆx(t))

−[(A(t, θ0)−K(t, θ0)C(t)) Υ(t) + Λ(t, σ_{X}(ˆx(t)))] ˜θ(t)
+Υ(t)

γΥ^{>}(t)C^{>}(t)Σ(t)(y(t)−C(t)ˆx(t))

=[A(t, θ0)−K(t, θ0)C(t)]η(t)
+Λ(t, σ_{X}(x(t))−σ_{X}(ˆx(t)))(θ−θ0).

(9) On the other hand, in view of (2) and sincex(t) =˜ η(t) + Υ(t)˜θ(t),

θ(t)˙˜ = −γΥ^{>}(t)C^{>}(t)Σ(t)C(t)˜x(t)

= −γΥ^{>}(t)C^{>}(t)Σ(t)C(t)

Υ(t)˜θ(t) +η(t)
.
(10)
The dynamics of the (η,θ)-system in (9), (10) is the˜
same as in (14)-(15) in [30] up to the perturbative term
Λ(t, σ_{X}(x(t))−σ_{X}(ˆx(t)))(θ−θ0)in (9). Because of the lat-
ter, we can no longer invoke cascade arguments to conclude
about the stability of the(η,θ)-system as in [30]. We will use˜
small-gain arguments instead, and the small-gain condition
will be enforced by imposing θ−θ0 to be small, which
explains why the convergence property we will guarantee is
local and not global (in general).

Let (ˆx,θ,¯Υ) be the solution to system (2) initialized at
(ˆx0,θ¯0,Υ0). We define P : R≥0×R^{n}^{θ} →R^{n}^{x}^{×n}^{x} as the
solution to−P˙(t, θ0) =P(t, θ0) [A(t, θ0)−K(t, θ0)C(t)]+

[A(t, θ_{0})−K(t, θ_{0})C(t)]^{>}P(t, θ_{0}) + I, with P(0, θ_{0}) =
P_{0} and P_{0} is symmetric and positive definite. Since θ_{0}
belongs to the closed ball B(θ, δ_{1}), Assumption 2 holds
and A(t, θ0)−K(t, θ0)C(t) is continuous and bounded,
Lemma 1 in [19] ensures that P(t, θ0) exists for allt≥0,
is continuously differentiable, bounded, symmetric, positive
definite and there exist a_{P}, aP >0, which are independent
of θ0, such that a_{P}I≤P(t, θ0) ≤aPI for anyt ≥0. We
defineV : (P, η)7→η^{>}P η. Let^{4}t≥0, in view of (9) and the
definition ofP(t, θ0)(we omit the time andθ0dependencies
of the solutions), V˙(P, η) = −||η||^{2}+ 2η^{>}PΛ(t, σX(x)−
σX(ˆx))(θ−θ0), where xˆ = x−η −Υ˜θ. The definition
of Λ(·,·) and the fact that the matrices A1(t), . . . , An_{θ}(t)
are bounded according to Assumption 1 imply that there
exists a constant a_{Λ} > 0 such that ||Λ(t, z)|| ≤ a_{Λ}||z||

for anyz ∈R^{n}^{x}. Consequently, since P ≤a_{P}I,V˙(P, η)≤

−||η||^{2}+2a_{P}a_{Λ}||η||·||σ_{X}(x)−σ_{X}(ˆx)||×||θ−θ_{0}||. From the
definition ofσ,||σ_{X}(x)−σ_{X}(ˆx)|| ≤n_{x}||x−x||ˆ =n_{x}||x|| ≤˜
nx||η||+nx||Υ|| · ||θ||. Thus,˜

V˙(P, η) ≤ −||η||^{2}+ 2nxaPaΛ||η||^{2}||θ−θ0||

+2nxaPaΛ||η|| · ||Υ|| · ||θ|| · ||θ˜ −θ0||

=−[1−2nxaPaΛ||θ−θ0||]||η||^{2}
+2nxaPaΛ||η|| · ||Υ|| · ||θ||˜

×||θ−θ0||.

(11)

4We can take anyt≥0here as the solutions we consider are defined for all positive times sincex(t)∈ X and in view of (2).

We now investigate theθ-system in (10). We need the next˜ claim for that purpose.

Claim 1: There exists a continuous function ν :
R^{n}^{x}^{×n}^{θ} →[1,∞), which is independent ofθ0, such that for
any continuous functionxˆ:R≥0→R^{n}^{x}, the corresponding
solution Υto the Υ-system in (2) initialized at Υ0 is such
that||Υ(t)|| ≤ν(Υ0)for anyt≥0.

Proof of Claim 1. The result follows from Assumption 2,
Lemma 1 in [19] and the fact thatΛ(t, σ_{X}(ˆx)) is bounded
uniformly with respect toθ0 ∈ B(θ,min{δ1, δ2}). The fact
that the image of ν is in [1,∞) can always be ensured by

adding1to any bound on ||Υ(t)||.

The matrixγΥ^{>}(t)C^{>}(t)Σ(t)C(t)Υ(t)is continuous and
bounded since so are C(t) by assumption, Σ(t) by de-
sign, and Υ(t) according to Claim 1 for the boundedness.

Therefore, since θ ∈ B(θ, δ_{2}), item (ii) of Theorem 1 is
assumed to hold, Σ(t) and C(t) are bounded and Claim 1
applies, according to Lemmas 1 and 5 in [19], the solu-
tionPe(t)to−˙

Pe(t) =−γPe(t)Υ^{>}(t)C^{>}(t)Σ(t)C(t)Υ(t)−
γΥ^{>}(t)C^{>}(t)Σ(t)C(t)Υ(t)>

Pe(t) +I, with P(0) =e Pe0

andPe_{0}is symmetric and positive definite, exists for allt≥0,
is continuously differentiable, bounded, symmetric, positive
definite and there exist a

Pe, a

Pe >0, which are independent of θ0, such thata

PeI≤Pe(t)≤a

PeI for any t≥0. Let Ve :
(P ,e θ)˜ 7→θ˜^{>}Peθ. Let˜ t≥0, in view of (10) and the definition
ofPe(t), ˙

Ve(P ,e θ) =˜ −||θ||˜ ^{2}−2γθ˜^{>}PeΥ^{>}(t)C^{>}(t)Σ(t)C(t)η.

By following similar steps as above, we derive

˙

Ve(P ,e θ)˜ ≤ −||θ||˜ ^{2}+ 2γa

PeaC||θ|| · ||Υ|| · ||η||,˜ (12)
whereaC >0 is a constant such that||C^{>}(t)Σ(t)C(t)|| ≤
aC for all t ≥ 0, which exists since C(t) and Σ(t) are
bounded.

DefineU : (P,P , η,e θ)˜ 7→V(P, η)+µVe(P ,e θ)˜ withµ >0 a constant, which will be selected below. In view of (11) and (12), for anyt≥0,

U(P,˙ P , η,e θ)˜ ≤ −[1−2n_{x}a_{P}a_{Λ}||θ−θ_{0}||]||η||^{2}
+2nxaPaΛ||Υ|| · ||θ|| · ||η|| · ||θ˜ −θ0||

−µ||θ||˜ ^{2}+ 2µγa

PeaC||θ|| · ||Υ|| · ||η||.˜ (13) According to Claim 1,||Υ|| ≤ν(Υ0), thus

U˙(P,P , η,e θ)˜ ≤ −[1−2nxaPaΛ||θ−θ0||]||η||^{2}
+2n_{x}a_{P}a_{Λ}ν(Υ_{0})||θ|| · ||η|| · ||θ˜ −θ_{0}||

−µ||θ||˜ ^{2}+ 2µγa

PeaCν(Υ0)||θ|| · ||η||˜

= −D

(||η||,||θ||),˜ M(||η||,||θ||)˜ E ,

(14) where M :=

M11 M12

? M22

with M11 := 1 −
2n_{x}a_{P}a_{Λ}||θ−θ_{0}||, M_{12} := −n_{x}a_{P}a_{Λ}ν(Υ_{0})||θ−θ_{0}|| −
µγa

Pea_{C}ν(Υ_{0})andM_{22}:=µ. MatrixMis positive definite
if and only if

1−2nxaPaΛ||θ−θ0||>0 [1−2nxaPaΛ||θ−θ0||]µ

−

nxaPaΛν(Υ0)||θ−θ0||+µγa_{P}_{e}aCν(Υ0)2

>0.

(15) The first inequality above is ensured as long as||θ−θ0||is sufficiently small, in particular we assume that||θ−θ0||<

1 4nxaPaΛ

so that 1−2n_{x}a_{P}a_{Λ}||θ−θ_{0}||> ^{1}_{2}. The second
inequality also holds with ||θ−θ0|| sufficiently small, by
takingµ < 1

2 γa

PeaCν(Υ0)2; note that the denominator is well-defined sinceν(Υ0)≥1in view of Claim 1. As a result, there existsδ∈(0,min{δ1, δ2}), which is independent ofθ0, such that||θ−θ0|| ≤δ implies the existence ofε >0 such that, fort≥0,

U˙(P,P , η,e θ)˜ ≤ −ε||(η,θ)||˜ ^{2}. (16)
Since min{a_{P}, µa

Pe}||(η,θ)||˜ ^{2} ≤U(P,P , η,e θ)˜ ≤(aP +
µaPe)||(η,θ)||˜ ^{2}, we deduce that (η(t),θ(t))˜ exponentially
converges to 0 as t → ∞. More precisely, there exist
c^{0}_{1}, c^{0}_{2} > 0, which depend on X,Mu,xˆ0,θ¯0,Υ0, such
that ||(η(t),θ(t))|| ≤˜ c^{0}_{1}e^{−c}^{0}^{2}^{t}||(η(0),θ(0)||. Moreover, since˜

||˜x(t)|| ≤ ||η(t)||+||Υ(t)|| · ||θ(t)|| ≤ ||η(t)||˜ +ν(Υ0)||θ(t)||˜
according to the definition ofηand Claim 1, we deduce that
there exist c1, c2 >0, which depend on X,Mu,xˆ0,θ¯0,Υ0,
such that ||(˜x(t),θ(t))|| ≤˜ c1e^{−c}^{2}^{t}||(˜x(0),θ(0))||, which˜
ends the proof.

C. WhenA(t, θ)is not affine in θ

We study the case where A(t, θ) is smooth in θ but not
necessarily affine inθ, as required in Assumption 1. Lett≥
0, θ ∈ R^{n}^{θ} and denote A(t, θ) = [aij(t, θ)](i,j)∈{1,...,nx}^{2}.
The Taylor expansion of each aij with respect to θ gives
a_{ij}(t, θ) = a_{ij}(t, θ_{0}) + ∂a_{ij}

∂θ (t, θ_{0})(θ−θ_{0}) +ρ_{ij}(t, θ, θ_{0})
where^{5} ρ_{ij}(t, θ, θ_{0}) =O(||θ−θ_{0}||^{2}). As a result,

A(t, θ) = A(t, θ0) +

n_{θ}

X

i=1

Ai(t, θ0)(θi−θ0,i) +R(t, θ, θ0),
(17)
where θ = (θ_{1}, . . . , θ_{n}_{θ}), θ_{0} = (θ_{0,1}, . . . , θ_{0,n}_{θ}),
Ak(t, θ0) := h_{∂a}

ij

∂θ_{k}(t, θ0)i

(i,j)∈{1,...,nx}^{2} for k ∈
{1, . . . , nθ},R(t, θ, θ0) := [ρij(t, θ, θ0)](i,j)∈{1,...,nx}^{2}.

In view of (17), we modify the observer in (2) as

˙ˆ

x(t) = A(t, θ0)ˆx(t) +B(t)u(t) + Λ(t, θ0, σ_{X}(ˆx(t)))¯θ(t)
+

K(t, θ0) +γΥ(t)Υ^{>}(t)C^{>}(t)Σ(t)

×(y(t)−C(t)ˆx(t))

θ(t) =˙¯ γΥ^{>}(t)C^{>}(t)Σ(t)(y(t)−C(t)ˆx(t))

Υ(t) = [A(t, θ˙ _{0})−K(t, θ_{0})C(t)] Υ(t) + Λ(t, θ_{0}, σ_{X}(ˆx(t)))
θ(t) = ¯ˆ θ(t) +θ_{0},

(18) where Λ(t, θ0, z) :=

A1(t, θ0)z . . . An_{θ}(t, θ0)z

for z ∈
R^{n}^{x}. The other matrices and parameters are selected as in
(2).

Proposition 1: Consider systems (1) and (18). Suppose the following holds.

(i) Assumptions 2 and 3 hold.

(ii) Item (ii) of Theorem 1 holds along the solutions to (1) and (18).

5We can write that ρij(t, θ, θ0) = O(||θ−θ0||^{2}) even though ρij

depends on the timet, sinceA(t, θ)is assumed to be bounded with respect to the time, see Section II.

There exist δ, c1, c2, c3 > 0, which depend on
X,Mu,xˆ_{0},θ¯_{0},Υ_{0} such that, if ||θ − θ_{0}|| ≤ δ, the
solution (ˆx,θ,¯Υ) to (18) initialized at (ˆx_{0},θ¯_{0},Υ_{0}) and
any solution to system (1) with x(0) ∈ X and input
u ∈ Mu are such that ||(x(t) − x(t),ˆ θ(t)ˆ − θ)|| ≤
c_{1}e^{−c}^{2}^{t}||(η(0),θ(0))||˜ +c_{3}||θ−θ_{0}||^{2}.
Sketch of proof. The proof follows the same steps as the
proof of Theorem 1. Lett≥0 andθ_{0}∈ B(θ,min{δ_{1}, δ_{2}})
where δ_{1}, δ_{2} come from Assumptions 2 and item (ii) of
Theorem 1, respectively. We first notice that instead of (6),
we have [A(t, θ)−A(t, θ0)]x(t) = Pnθ

i=1Ai(t, θ0)(θi − θ0,i)x(t) + R(t, θ, θ0)x(t) = Λ(t, θ0, x(t))(θ − θ0) + R(t, θ, θ0)x(t)in view of (17) and the definition ofΛ. Thus, by following the proof of Theorem 1, we obtain the next equation instead of (9) fort≥0,x(t)∈ X withu∈ Mu,

˙

η(t) = [A(t, θ0)−K(t, θ0)C(t)]η(t)

+Λ(t, θ_{0}, σ_{X}(x(t))−σ_{X}(ˆx(t)))(θ−θ_{0})
+R(t, θ, θ0)x(t),

(19) and (10) still holds.

Letxbelong toSX with input u∈ Mu, and(ˆx,θ,¯Υ) be the solution to system (2) initialized at(ˆx0,θ¯0,Υ0). Lett≥ 0. The same Lyapunov analysis as in the proof of Theorem 1, leads to the next equation instead of (16) when||θ−θ0|| ≤δ for someδ >0

U˙(P,P , η,e θ)˜ ≤ −ε||(η,θ)||˜ ^{2}+ 2η^{T}P(t, θ0)R(t, θ, θ0)x.

(20)
Since P(t, θ0) ≤ aPI, R(t, θ, θ0) =
[ρij(t, θ, θ0)](i,j)∈{1,...,nx}^{2}andρij(t, θ, θ0) =O(||θ−θ0||^{2}),
andx(t)∈ X according to Assumption 3, there existsa≥0
such that2||P(t, θ0)R(t, θ, θ0)x|| ≤a||θ−θ0||^{2}. Hence,

U˙(P,P , η,e θ)˜ ≤ −ε||(η,θ)||˜ ^{2}+a||η|| · ||θ−θ_{0}||^{2}

≤ −ε||(η,θ)||˜ ^{2}+a||(η,θ)|| · ||θ˜ −θ0||^{2}
(21)
from which we deduce, using a||(η,θ)|| · ||θ˜ −θ0||^{2} ≤

ε

2||(η,θ)||˜ ^{2}+^{2}_{ε}a^{2}||θ−θ0||^{4},

U˙(P,P , η,e θ)˜ ≤ −^{ε}_{2}||(η,θ)||˜ ^{2}+^{2}_{ε}a^{2}||θ−θ_{0}||^{4}. (22)
Since min{a_{P}, µa

Pe}||(η,θ)||˜ ^{2} ≤ U(P,P , η,e θ)˜ ≤
(aP +µa

Pe)||(η,θ)||˜ ^{2}, we deduce from (22) by applying
the comparison lemma (see Lemma 3.4 in [15]) that

||(η(t),θ(t))|| ≤˜ c1e^{−c}^{2}^{t}||(η(0),θ(0))||˜ +c3||θ−θ0||^{2} for
some c1, c2, c3 > 0. The desired result for x˜ then follows
by notingx˜ =η+ Υ˜θ and thatΥis bounded according to

Claim 1.

As already mentioned, Assumption 2 implies that we know
a state observer for system (1) when θ = θ_{0}. If we would
implement this classical (non-adaptive) observer on system
(1) withθ6=θ0, the state estimate would converge toxup to
an error of the order of||θ−θ0||, and, trivially, the parameter
estimation error would beθ−θ0. Proposition 1 shows that
these properties can be improved by employing the adaptive
observer (18), which provides estimates with errors of the
order of||θ−θ0||^{2}, after a sufficiently long time. We are not

Fig. 1. Coupled mass-spring system.

able to ensure the asymptotic convergence of the estimates of the true values, contrary to Section III-A, because of the perturbative termR(t, θ, θ0)in (17).

IV. ILLUSTRATIVE EXAMPLE

Consider the mass-spring system with two elements de-
picted in Figure 1. The system is be modeled by (1)
with: x = (x1, x2, x3, x4) ∈ R^{4}, where x1, x3 are the
displacements of the first and the second mass from their
equilibrium and x2, x4 are the velocity of the first and the
second mass, respectively; u is in the input applied to the
second mass;y= (x_{1}, x_{2})∈R^{2}, which means that only the
variables of the first mass are measured. The state matrix
isA=

0 1 0 0

−^{k}_{m}^{1}^{+θ}

1 −_{m}^{b}^{1}

1 θ

m_{1} 0

0 0 0 1

θ

m_{2} 0 −^{k}^{2}_{m}^{+θ}

2 −_{m}^{b}^{2}

2

,m_{1}= 100 and
m2 = 100 are the masses, k1 = 5 and k2 = 10 are the
spring stiffness as shown in Figure 1, θ >0 is the stiffness
of the middle spring, which is assumed to be unknown and
b1 = 0.1 andb2= 0.4 are the damping coefficients. Matrix
B is (0 0 0_{m}^{1}

2)^{>} and C =

1 0 0 0 0 1 0 0

. Assumption

1 holds with A0 =

0 1 0 0

−_{m}^{k}^{1}

1 −_{m}^{b}^{1}

1 0 0

0 0 0 1

0 0 −_{m}^{k}^{2}

2 −_{m}^{b}^{2}

2

and

A1 =

0 0 0 0

−_{m}^{1}

1 0 _{m}^{1}

1 0

0 0 0 0

1

m_{2} 0 −_{m}^{1}

2 0

. Assumption 2 is verified as(A, C)is observable as θ6= 0. Takingu(t) = 10 sin(10t) for any t ≥ 0, Assumption 3 is satisfied since the infinity norm of this input is bounded andAis Hurwitz.

We have designed the adaptive observer as in (2) with
γ = 10^{3}, Σ(t) = 10^{3} and K such that the eigenvalues of
A−KC are(−1,−1.5,−2,−2.5). We have run simulations
withθ= 15, and we have considered different values ofθ0,
x(0),x(0),ˆ θ(0)¯ andΥ(0). Figures 2 and 3 show that the state
and parameter estimates do track the state and parameter of
system (1), respectively, whenθ_{0} = 20, x(0) = (1,0,2,0),
ˆ

x(0) = (0,0,0,0), θ(0) = 0¯ and Υ_{0} = 0. We have then
varied the latter. Simulations suggest that the convergence
of the adaptive observer may be independent of the initial
conditions of x, ˆx, θ¯ and Υ: only θ0 seems to matter.

The asymptotic convergence of the estimation errors is seen wheneverθ0∈[θ−11, θ+ 75]. This means thatθ0 does not need to be very close toθfor the adaptive observer to work for this example.

Fig. 2. Norm of the state estimation error.

Fig. 3. Parameterθ,θ0and the estimateθ.ˆ

V. CONCLUSION

We presented an adaptive observer for linear time-varying system whose state matrix A(t, θ) depends on unknown parameters. WhenA(t, θ)is affine inθ, the proposed scheme ensures the exponential convergence of the estimates to the true values provided some initial guess of the unknown parameter is sufficiently closed to the latter and a persis- tence of excitation holds. When A(t, θ) is only smooth in θ, a modified version of the observer has been proposed, which ensures the approximate convergence to zero of the estimation errors.

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